\documentclass[9pt]{article}
\usepackage{amsmath}
\begin{document}
Read in the following dictionary:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  0.0 & -1.00 x_{1} & -2.00 x_{2} & +  9.00 x_{3} & +  6.00 x_{4} & -8.00 x_{5} & -5.00 x_{6}\\
 x_{8}   &  -2.0  &   & -2.00 x_{2} & +  8.00 x_{3} & -6.00 x_{4} & -5.00 x_{5} & -5.00 x_{6}\\
 x_{9}   &  2.0 & +  5.00 x_{1} & -9.00 x_{2} & + 10.00 x_{3} & -8.00 x_{4} & +  9.00 x_{5} & +  9.00 x_{6}\\
 x_{10}   &  -2.0 & -8.00 x_{1} & -2.00 x_{2} & +  6.00 x_{3} & +  1.00 x_{4} & +  5.00 x_{5} & +  6.00 x_{6}\\
 x_{11}   &  1.0 & +  9.00 x_{1} & +  3.00 x_{2} & -4.00 x_{3} & +  7.00 x_{4} & +  8.00 x_{5} & +  2.00 x_{6}\\
 x_{12}   &  0.0 & -3.00 x_{1} & +  2.00 x_{2} & +  9.00 x_{3} & +  9.00 x_{4} & +  5.00 x_{5} & +  2.00 x_{6}\\
\hline
z    &  0.0 & +  1.00 x_{1} & +  3.00 x_{2} & +  2.00 x_{3} & +  5.00 x_{4} & -3.00 x_{5} & -3.00 x_{6}\\
\end{array}\]
\subsection{Initialization Phase: Dual Problem Solving}
New Objective in primal was changed to : \[ \max\ \sum_{j=1}^{6}\ - x_j \] 
Primal variable $x_j$ corresponds to dual variable $y_j$ for $j = 1,\ldots,12$
Dual Dictionary (with objective changed is): 
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  1.0 & +  1.00 y_{7} &   & -5.00 y_{9} & +  8.00 y_{10} & -9.00 y_{11} & +  3.00 y_{12}\\
 y_{2}   &  1.0 & +  2.00 y_{7} & +  2.00 y_{8} & +  9.00 y_{9} & +  2.00 y_{10} & -3.00 y_{11} & -2.00 y_{12}\\
 y_{3}   &  1.0 & -9.00 y_{7} & -8.00 y_{8} & -10.00 y_{9} & -6.00 y_{10} & +  4.00 y_{11} & -9.00 y_{12}\\
 y_{4}   &  1.0 & -6.00 y_{7} & +  6.00 y_{8} & +  8.00 y_{9} & -1.00 y_{10} & -7.00 y_{11} & -9.00 y_{12}\\
 y_{5}   &  1.0 & +  8.00 y_{7} & +  5.00 y_{8} & -9.00 y_{9} & -5.00 y_{10} & -8.00 y_{11} & -5.00 y_{12}\\
 y_{6}   &  1.0 & +  5.00 y_{7} & +  5.00 y_{8} & -9.00 y_{9} & -6.00 y_{10} & -2.00 y_{11} & -2.00 y_{12}\\
\hline
z    &  -0  &   & +  2.00 y_{8} & -2.00 y_{9} & +  2.00 y_{10} & -1.00 y_{11} &   \\
\end{array}\]
Initialization succeeded in finding final dual dictionary with 4 pivots
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 y_{1}   &  2.09267241379 & -3.75 y_{7} & -0.42 y_{3} & -15.25 y_{9} & -2.12 y_{6} & +  1.44 y_{5} & +  2.20 y_{12}\\
 y_{2}   &  1.27801724138 & -1.26 y_{7} & -0.26 y_{3} & +  6.24 y_{9} & -0.35 y_{6} & +  0.33 y_{5} & -3.39 y_{12}\\
 y_{8}   &  0.0129310344828 & -0.85 y_{7} & -0.08 y_{3} & -0.20 y_{9} & +  0.15 y_{6} & -0.08 y_{5} & -0.83 y_{12}\\
 y_{4}   &  0.713362068966 & -14.51 y_{7} & -0.35 y_{3} & + 10.23 y_{9} & -0.08 y_{6} & +  0.72 y_{5} & -8.73 y_{12}\\
 y_{11}   &  0.0280172413793 & +  0.49 y_{7} & -0.01 y_{3} & -0.26 y_{9} & +  0.15 y_{6} & -0.17 y_{5} & -0.64 y_{12}\\
 y_{10}   &  0.168103448276 & -0.04 y_{7} & -0.06 y_{3} & -1.58 y_{9} & -0.09 y_{6} & -0.01 y_{5} & -0.81 y_{12}\\
\hline
z    &  0.334051724138 & -2.27 y_{7} & -0.28 y_{3} & -5.29 y_{9} & -0.05 y_{6} & -0.00 y_{5} & -2.66 y_{12}\\
\end{array}\]
Primal Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.26939655172 & +  3.75 x_{1} & +  1.26 x_{2} & +  0.85 x_{8} & + 14.51 x_{4} & -0.49 x_{11} & +  0.04 x_{10}\\
 x_{3}   &  0.282327586207 & +  0.42 x_{1} & +  0.26 x_{2} & +  0.08 x_{8} & +  0.35 x_{4} & +  0.01 x_{11} & +  0.06 x_{10}\\
 x_{9}   &  5.28879310345 & + 15.25 x_{1} & -6.24 x_{2} & +  0.20 x_{8} & -10.23 x_{4} & +  0.26 x_{11} & +  1.58 x_{10}\\
 x_{6}   &  0.0474137931034 & +  2.12 x_{1} & +  0.35 x_{2} & -0.15 x_{8} & +  0.08 x_{4} & -0.15 x_{11} & +  0.09 x_{10}\\
 x_{5}   &  0.00431034482759 & -1.44 x_{1} & -0.33 x_{2} & +  0.08 x_{8} & -0.72 x_{4} & +  0.17 x_{11} & +  0.01 x_{10}\\
 x_{12}   &  2.65732758621 & -2.20 x_{1} & +  3.39 x_{2} & +  0.83 x_{8} & +  8.73 x_{4} & +  0.64 x_{11} & +  0.81 x_{10}\\
\hline
z    &  -0.334051724138 & -2.09 x_{1} & -1.28 x_{2} & -0.01 x_{8} & -0.71 x_{4} & -0.03 x_{11} & -0.17 x_{10}\\
\end{array}\]
Primal Dictionary with original objective is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.26939655172 & +  3.75 x_{1} & +  1.26 x_{2} & +  0.85 x_{8} & + 14.51 x_{4} & -0.49 x_{11} & +  0.04 x_{10}\\
 x_{3}   &  0.282327586207 & +  0.42 x_{1} & +  0.26 x_{2} & +  0.08 x_{8} & +  0.35 x_{4} & +  0.01 x_{11} & +  0.06 x_{10}\\
 x_{9}   &  5.28879310345 & + 15.25 x_{1} & -6.24 x_{2} & +  0.20 x_{8} & -10.23 x_{4} & +  0.26 x_{11} & +  1.58 x_{10}\\
 x_{6}   &  0.0474137931034 & +  2.12 x_{1} & +  0.35 x_{2} & -0.15 x_{8} & +  0.08 x_{4} & -0.15 x_{11} & +  0.09 x_{10}\\
 x_{5}   &  0.00431034482759 & -1.44 x_{1} & -0.33 x_{2} & +  0.08 x_{8} & -0.72 x_{4} & +  0.17 x_{11} & +  0.01 x_{10}\\
 x_{12}   &  2.65732758621 & -2.20 x_{1} & +  3.39 x_{2} & +  0.83 x_{8} & +  8.73 x_{4} & +  0.64 x_{11} & +  0.81 x_{10}\\
\hline
z    &  0.409482758621 & -0.18 x_{1} & +  3.47 x_{2} & +  0.37 x_{8} & +  7.62 x_{4} & -0.03 x_{11} & -0.18 x_{10}\\
\end{array}\]
\section{Optimization Phase Simplex}
Starting Dictionary is:
\[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.26939655172 & +  3.75 x_{1} & +  1.26 x_{2} & +  0.85 x_{8} & + 14.51 x_{4} & -0.49 x_{11} & +  0.04 x_{10}\\
 x_{3}   &  0.282327586207 & +  0.42 x_{1} & +  0.26 x_{2} & +  0.08 x_{8} & +  0.35 x_{4} & +  0.01 x_{11} & +  0.06 x_{10}\\
 x_{9}   &  5.28879310345 & + 15.25 x_{1} & -6.24 x_{2} & +  0.20 x_{8} & -10.23 x_{4} & +  0.26 x_{11} & +  1.58 x_{10}\\
 x_{6}   &  0.0474137931034 & +  2.12 x_{1} & +  0.35 x_{2} & -0.15 x_{8} & +  0.08 x_{4} & -0.15 x_{11} & +  0.09 x_{10}\\
 x_{5}   &  0.00431034482759 & -1.44 x_{1} & -0.33 x_{2} & +  0.08 x_{8} & -0.72 x_{4} & +  0.17 x_{11} & +  0.01 x_{10}\\
 x_{12}   &  2.65732758621 & -2.20 x_{1} & +  3.39 x_{2} & +  0.83 x_{8} & +  8.73 x_{4} & +  0.64 x_{11} & +  0.81 x_{10}\\
\hline
z    &  0.409482758621 & -0.18 x_{1} & +  3.47 x_{2} & +  0.37 x_{8} & +  7.62 x_{4} & -0.03 x_{11} & -0.18 x_{10}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{5} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.28571428571 & -1.71 x_{1} & -3.79 x_{5} & +  1.14 x_{8} & + 11.79 x_{4} & +  0.14 x_{11} & +  0.07 x_{10}\\
 x_{3}   &  0.285714285714 & -0.71 x_{1} & -0.79 x_{5} & +  0.14 x_{8} & -0.21 x_{4} & +  0.14 x_{11} & +  0.07 x_{10}\\
 x_{9}   &  5.20779220779 & + 42.39 x_{1} & + 18.79 x_{5} & -1.26 x_{8} & +  3.30 x_{4} & -2.90 x_{11} & +  1.42 x_{10}\\
 x_{6}   &  0.0519480519481 & +  0.60 x_{1} & -1.05 x_{5} & -0.06 x_{8} & -0.68 x_{4} & +  0.03 x_{11} & +  0.10 x_{10}\\
 x_{2}   &  0.012987012987 & -4.35 x_{1} & -3.01 x_{5} & +  0.23 x_{8} & -2.17 x_{4} & +  0.51 x_{11} & +  0.03 x_{10}\\
 x_{12}   &  2.7012987013 & -16.94 x_{1} & -10.20 x_{5} & +  1.62 x_{8} & +  1.38 x_{4} & +  2.35 x_{11} & +  0.90 x_{10}\\
\hline
z    &  0.454545454545 & -15.27 x_{1} & -10.45 x_{5} & +  1.18 x_{8} & +  0.09 x_{4} & +  1.73 x_{11} & -0.09 x_{10}\\
\end{array}\]


 $ x_{4} $ enters and $ x_{2} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  2.35628742515 & -25.36 x_{1} & -20.16 x_{5} & +  2.41 x_{8} & -5.43 x_{2} & +  2.90 x_{11} & +  0.21 x_{10}\\
 x_{3}   &  0.284431137725 & -0.28 x_{1} & -0.49 x_{5} & +  0.12 x_{8} & +  0.10 x_{2} & +  0.09 x_{11} & +  0.07 x_{10}\\
 x_{9}   &  5.22754491018 & + 35.77 x_{1} & + 14.21 x_{5} & -0.90 x_{8} & -1.52 x_{2} & -2.13 x_{11} & +  1.46 x_{10}\\
 x_{6}   &  0.0479041916168 & +  1.95 x_{1} & -0.11 x_{5} & -0.14 x_{8} & +  0.31 x_{2} & -0.13 x_{11} & +  0.10 x_{10}\\
 x_{4}   &  0.0059880239521 & -2.01 x_{1} & -1.39 x_{5} & +  0.11 x_{8} & -0.46 x_{2} & +  0.23 x_{11} & +  0.01 x_{10}\\
 x_{12}   &  2.70958083832 & -19.71 x_{1} & -12.12 x_{5} & +  1.77 x_{8} & -0.64 x_{2} & +  2.67 x_{11} & +  0.92 x_{10}\\
\hline
z    &  0.455089820359 & -15.46 x_{1} & -10.58 x_{5} & +  1.19 x_{8} & -0.04 x_{2} & +  1.75 x_{11} & -0.09 x_{10}\\
\end{array}\]


 $ x_{8} $ enters and $ x_{6} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  3.19565217391 & +  8.85 x_{1} & -22.15 x_{5} & -17.52 x_{6} & +  0.02 x_{2} & +  0.59 x_{11} & +  1.89 x_{10}\\
 x_{3}   &  0.326086956522 & +  1.41 x_{1} & -0.59 x_{5} & -0.87 x_{6} & +  0.37 x_{2} & -0.02 x_{11} & +  0.15 x_{10}\\
 x_{9}   &  4.91304347826 & + 22.96 x_{1} & + 14.96 x_{5} & +  6.57 x_{6} & -3.57 x_{2} & -1.26 x_{11} & +  0.83 x_{10}\\
 x_{8}   &  0.347826086957 & + 14.17 x_{1} & -0.83 x_{5} & -7.26 x_{6} & +  2.26 x_{2} & -0.96 x_{11} & +  0.70 x_{10}\\
 x_{4}   &  0.0434782608696 & -0.48 x_{1} & -1.48 x_{5} & -0.78 x_{6} & -0.22 x_{2} & +  0.13 x_{11} & +  0.09 x_{10}\\
 x_{12}   &  3.32608695652 & +  5.41 x_{1} & -13.59 x_{5} & -12.87 x_{6} & +  3.37 x_{2} & +  0.98 x_{11} & +  2.15 x_{10}\\
\hline
z    &  0.869565217391 & +  1.43 x_{1} & -11.57 x_{5} & -8.65 x_{6} & +  2.65 x_{2} & +  0.61 x_{11} & +  0.74 x_{10}\\
\end{array}\]


 $ x_{1} $ enters and $ x_{4} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  4.0 & -18.50 x_{4} & -49.50 x_{5} & -32.00 x_{6} & -4.00 x_{2} & +  3.00 x_{11} & +  3.50 x_{10}\\
 x_{3}   &  0.454545454545 & -2.95 x_{4} & -4.95 x_{5} & -3.18 x_{6} & -0.27 x_{2} & +  0.36 x_{11} & +  0.41 x_{10}\\
 x_{9}   &  7.0 & -48.00 x_{4} & -56.00 x_{5} & -31.00 x_{6} & -14.00 x_{2} & +  5.00 x_{11} & +  5.00 x_{10}\\
 x_{8}   &  1.63636363636 & -29.64 x_{4} & -44.64 x_{5} & -30.45 x_{6} & -4.18 x_{2} & +  2.91 x_{11} & +  3.27 x_{10}\\
 x_{1}   &  0.0909090909091 & -2.09 x_{4} & -3.09 x_{5} & -1.64 x_{6} & -0.45 x_{2} & +  0.27 x_{11} & +  0.18 x_{10}\\
 x_{12}   &  3.81818181818 & -11.32 x_{4} & -30.32 x_{5} & -21.73 x_{6} & +  0.91 x_{2} & +  2.45 x_{11} & +  3.14 x_{10}\\
\hline
z    &  1.0 & -3.00 x_{4} & -16.00 x_{5} & -11.00 x_{6} & +  2.00 x_{2} & +  1.00 x_{11} & +  1.00 x_{10}\\
\end{array}\]


 $ x_{2} $ enters and $ x_{1} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  3.2 & -0.10 x_{4} & -22.30 x_{5} & -17.60 x_{6} & +  8.80 x_{1} & +  0.60 x_{11} & +  1.90 x_{10}\\
 x_{3}   &  0.4 & -1.70 x_{4} & -3.10 x_{5} & -2.20 x_{6} & +  0.60 x_{1} & +  0.20 x_{11} & +  0.30 x_{10}\\
 x_{9}   &  4.2 & + 16.40 x_{4} & + 39.20 x_{5} & + 19.40 x_{6} & + 30.80 x_{1} & -3.40 x_{11} & -0.60 x_{10}\\
 x_{8}   &  0.8 & -10.40 x_{4} & -16.20 x_{5} & -15.40 x_{6} & +  9.20 x_{1} & +  0.40 x_{11} & +  1.60 x_{10}\\
 x_{2}   &  0.2 & -4.60 x_{4} & -6.80 x_{5} & -3.60 x_{6} & -2.20 x_{1} & +  0.60 x_{11} & +  0.40 x_{10}\\
 x_{12}   &  4.0 & -15.50 x_{4} & -36.50 x_{5} & -25.00 x_{6} & -2.00 x_{1} & +  3.00 x_{11} & +  3.50 x_{10}\\
\hline
z    &  1.4 & -12.20 x_{4} & -29.60 x_{5} & -18.20 x_{6} & -4.40 x_{1} & +  2.20 x_{11} & +  1.80 x_{10}\\
\end{array}\]


 $ x_{10} $ enters and $ x_{9} $ leaves 

 \[\begin{array}{c| c c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} c@{\hskip 2pt} }
 x_{7}   &  16.5 & + 51.83 x_{4} & + 101.83 x_{5} & + 43.83 x_{6} & + 106.33 x_{1} & -10.17 x_{11} & -3.17 x_{9}\\
 x_{3}   &  2.5 & +  6.50 x_{4} & + 16.50 x_{5} & +  7.50 x_{6} & + 16.00 x_{1} & -1.50 x_{11} & -0.50 x_{9}\\
 x_{10}   &  7.0 & + 27.33 x_{4} & + 65.33 x_{5} & + 32.33 x_{6} & + 51.33 x_{1} & -5.67 x_{11} & -1.67 x_{9}\\
 x_{8}   &  12.0 & + 33.33 x_{4} & + 88.33 x_{5} & + 36.33 x_{6} & + 91.33 x_{1} & -8.67 x_{11} & -2.67 x_{9}\\
 x_{2}   &  3.0 & +  6.33 x_{4} & + 19.33 x_{5} & +  9.33 x_{6} & + 18.33 x_{1} & -1.67 x_{11} & -0.67 x_{9}\\
 x_{12}   &  28.5 & + 80.17 x_{4} & + 192.17 x_{5} & + 88.17 x_{6} & + 177.67 x_{1} & -16.83 x_{11} & -5.83 x_{9}\\
\hline
z    &  14.0 & + 37.00 x_{4} & + 88.00 x_{5} & + 40.00 x_{6} & + 88.00 x_{1} & -8.00 x_{11} & -3.00 x_{9}\\
\end{array}\]


 $ x_{1} $ enters and Unbounded Dictionary!


 $ x_{1} $ enters and Unbounded Dictionary!
\end{document}
